Question

Find the time period of small oscillations of the following systems. (a) A metre stick suspended through the 20 cm mark. (b) A ring of mass m and radius r suspended through a point on its periphery. (c) A uniform square plate of edge a suspended through a corner. (d) A uniform disc of mass m and radius r suspended through a point $\frac{r}{2}$ away from the centre.

Answer 1

(a) M.I. about the pt. $a=1=I.C.G.+m{h}^2$

$=\frac{M{i}^2}{12}+m{h}^2$

$=\frac{M{i}^2}{12}+M(0.3{)}^2$

$=M(\frac{1}{2}+0.09)$

$=M\left(\frac{1+1.08}{12}\right)$

$=M\left(\frac{2.08}{12}\right)$

$T=2x\sqrt{\frac{l}{mgl}}$

$=2\pi \sqrt{\frac{2.08m}{m\times 9.8\times 0.3}}$

(1 = dist. between C.G. and pt. of suspension)

(b) Moment of inertia about A,

$I=I.C.G.+m{h}^2+m{r}^2$

$=2m{r}^2$

$\therefore$ Time period $=2\pi \sqrt{\frac{I}{mgl}}$

$=2\pi \sqrt{\frac{2m{r}^2}{mgr}}=2\pi \sqrt{\frac{2r}{g}}$

(c) $l_{xx}$ (corner) $=\left(\frac{a^2+a^2}{3}\right)=\frac{2m}{3}{a}^2$

In the $\Delta ABC,l^2+l^2=a^2$

$\therefore l=\frac{a}{\sqrt{2}}$

$\therefore T=2\pi \sqrt{\frac{I}{mgl}}$

$=2\pi \sqrt{\frac{2m{a}^2}{3mgl}}$

$=2\pi \sqrt{\frac{2a^2\sqrt{2}}{3ga}}$

$=2\pi \sqrt{\frac{\sqrt{8a}}{3g}}$

(d) $h=\frac{r}{2},$

$l=\frac{r}{2}$ = Dist. between C.G. and suspension point

M.I., about A, $l-I_{C.G.+M{h}^2}$

$=\frac{m{r}^2}{2}+m{\left(\frac{r}{2}\right)}^2$

$=m{r}^2(\frac{1}{2}+\frac{1}{4})=\frac{3}{4}m{r}^2$

$\therefore T=2\pi \sqrt{\frac{I}{mgl}}=2\pi \sqrt{\frac{3m{r}^2}{4mgl}}$

$=2\pi \sqrt{\frac{3r^2}{4gr/2}}=2\pi \sqrt{\frac{3r}{2g}}$